Based on my experience in Italy, Lufthansa offices still offer free copies at the ticketing counter. I’m not too sure how many people buy tickets at the airport nowadays, let alone do so with a paper timetable, but they are still there! :-)
Many people. Mostly business people who need to be somewhere on short notice. Until very recently, my wife was in this group.
Or does the airport ticketing desk have a later cutoff time?
In her situation, no. The phone would ring at 5am, she'd get dressed and grab her "go" bag and walk down to the subway, then buys her ticket (usually one-way) at the airport counter.
In theory she could try to book tickets on her phone on the train, but connectivity can be iffy on the train, and buying an expensive airline ticket is not the sort of thing you want to take unnecessary chances with. (Think double-purchases, false purchases, etc...)